Talk:PlanetPhysics/Geiger's Method

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Geiger's Method %%% Primary Category Code: 91.30.Px %%% Filename: GeigersMethod.tex %%% Version: 2 %%% Owner: aerringt %%% Author(s): aerringt %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Geiger's method \cite{Geiger} is an iterative procedure using Gauss-Newton optimization to determine the location of an earthquake, or seismic event. Originally his method was developed to obtain the origin time and \htmladdnormallink{Epicentre}{http://planetphysics.us/encyclopedia/Epicentre.html} but it is easily extended to include the \htmladdnormallink{Focal Depth}{http://planetphysics.us/encyclopedia/FocalDepth.html} for \htmladdnormallink{Hypocentre}{http://planetphysics.us/encyclopedia/Hypocenter.html} determination.

Given a set of $M$ arrival times $t_i$ find the origin time $t_0$ and the hypocentre in cartesian coordinatios $(x_0,y_0,z_0)$ which minimize the objective \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} \begin{equation} F(\mathbf{X})=\sum_{i=1}^{M}r_i^2. \end{equation} Here, $r_i$ is the difference between observed and calculated arrival times \begin{equation} r_i=t_i-t_0-T_i, \end{equation} and the unknown \htmladdnormallink{parameter}{http://planetphysics.us/encyclopedia/Parameter.html} \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} is \begin{equation} \mathbf{X}=(t_0,x_0,y_0,z_0)^{\mathrm{T}} \end{equation} In \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} form (1) becomes \begin{equation} F(\mathbf{X})=\mathbf{r}^{\mathrm{T}}\mathbf{r} \end{equation} The Gauss--Newton procedure requires an initial guess of the sought parameters, denoted here as \begin{equation} \mathbf{X}^*=(t_0^*,x_0^*,y_0^*,z_0^*)^{\mathbf{T}}, \end{equation} which are then used to calculate the adjustment vector \begin{equation} \delta\mathbf{X}=(\delta t_0,\delta x_0,\delta y_0,\delta z_0)^{\mathrm{T}} \end{equation} in \begin{equation} \label{big_matrix} \mathbf{A}^{\mathrm{T}}\mathbf{A}\delta\mathbf{X}=-\mathbf{A}^{\mathrm{T}}\mathbf{r}. \end{equation} The Jacobian matrix $\mathbf{A}$ is defined as \begin{equation} \mathbf{A}=\left( \begin{array}{cccc} \partial r_1/\partial t_0 & \partial r_1/ \partial x_0 & \partial r_1/\partial y_0 & \partial r_1/\partial z_0 \\

\partial r_2/\partial t_0 & \partial r_2/ \partial x_0 & \partial r_2/\partial y_0 & \partial r_2/\partial z_0 \\

\vdots & \vdots & \vdots & \vdots \\

\partial r_M/\partial t_0 & \partial r_M/ \partial x_0 & \partial r_M/\partial y_0 & \partial r_M/\partial z_0 \\ \end{array}\right). \end{equation} The partial derivatives are evaluated at the initial guess, or trial vector, $\mathbf{X}^*$. Equation (\ref{big_matrix}) can be rewritten as \begin{equation} \label{final_matrix} \mathbf{G}\delta\mathbf{X}=\mathbf{g}. \end{equation} Using (\ref{final_matrix}) and an initial guess $\mathbf{X}^*$ an adjustment vector can be calculated. The initial guess can then be updated $\mathbf{X}^*+\delta \mathbf{X}$ and used as the inital guess in the next run of the \htmladdnormallink{algorithm}{http://planetphysics.us/encyclopedia/RecursiveFunction.html}. In this manner the sought parameters $\mathbf{X}$ can be determined to some tolerance.

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\bibitem{Geiger}Geiger, L., Probability method for the determination of earthquake epicenters from the arrival time only. {\it Bull. St. Louis Univ.} vol. 8, pp. 60-71. \bibitem{Lee}Lee, W. H. K. and Stewart, S. W. {\it Principles and Applications of Microearthquake Networks,} Academic Press, New York. 1981

\bibitem{Gib}Gibowicz, S. J. and Kijko, A. {\it An Introduction to Mining Seismology,} Academic Press, New York. 1994.

\end{thebibliography}

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